Connecting the Endpoints of 2 Tubes while only moving one
I'm illustrating this with triangular openings, because it's easier to see. We're going to move green to red. All moves in edit mode. Pivot set to '3D Cursor'


Starting with 2 skew open shapes. Snap set to 'Vertex', 'Active'. Create a new Custom Orientation : Ctrl Alt Space from 3 vertices selected on the target opening.





With all vertices of green selected, and a suitable vertex made active, free translate G and snap the active vertex to the corresponding vertex in red. These points will not move any more. Put the 3D cursor there: ShiftS > Cursor to Active





Free rotate green around until it's somewhere in the vicinity. This is only to make things easier to see.





Now we need a view straight across the plane of the opening in red. With 3 verts in the red opening selected, try the orthogonal views of them (ShiftNumpad1,3or7 and the CtrlShift variants) until you reach a view suitable for the next steps. Set snap to 'Edge', 'Active'. 





Make a suitable vertex of green active, and rotate all in screen space, snapping the vertex to an edge of red in this view. We now have an edge in the right plane...





...but not aligned to its corresponding edge. So we now rotate all in the Custom orientation we created earlier , RZZ setting snap back to 'Vertex', 'Active'.





Now we have an edge in place. We want to rotate around it. Create a new Custom Orientation from the edge....





...and rotate around it, having made a suitable green vertex active.. (RYY)





...so it can snap to its partner.